de Broglie Wavelength Calculator

Calculate the de Broglie wavelength of any particle from its mass and velocity.

Electron: 9.109×10⁻³¹ kg; Proton: 1.673×10⁻²⁷ kg
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Matter Waves and Momentum

The de Broglie wavelength says that momentum has a wave scale: every material particle can be associated with λ=h/p. Planck's constant h is extremely small, so everyday objects have wavelengths far below measurable scales. Electrons, neutrons, atoms, and other microscopic particles can have wavelengths comparable to atomic spacings, which is why diffraction and interference can reveal their wave behavior.

For a nonrelativistic particle, momentum is p=mv, giving λ=h/(mv). This form shows why increasing either mass or speed shortens the wavelength. The distinction students often miss is that h/(mv) is an approximation when speed becomes a significant fraction of the speed of light. Relativistically, momentum is p=γmv, so the wavelength is shorter than the nonrelativistic estimate by the Lorentz factor γ.

λ=h/p   and, for v«c,   λ=h/(mv)
SymbolMeaningWhy it appears / units
λde Broglie wavelengthm; the spatial wave scale associated with the particle.
hPlanck constant6.62607015×10−34J·s.
pMomentumkg·m/s; wavelength is inversely proportional to momentum.
mRest masskg; used in p=mv only in the nonrelativistic regime.
vParticle speedm/s; greater speed means greater momentum and shorter wavelength.

Wave behavior becomes easiest to observe when the wavelength is comparable to a feature in the experiment, such as crystal-plane spacing or a narrow aperture. A tiny wavelength does not mean wave-particle duality disappears; it means the interference structure becomes too fine to detect with ordinary apparatus.

Worked Examples

Example 1: Electron at 1×10⁶ m/s
m=9.109e-31 kg, v=1e6 m/s
Result: 7.274×10⁻¹⁰ m (0.727 nm)
X-ray range — explains electron diffraction
Example 2: Proton at 1×10⁶ m/s
m=1.673e-27 kg, v=1e6 m/s
Result: 3.965×10⁻¹³ m
Nuclear scale wavelength
Example 3: Thermal-scale neutron
m=1.675×10−27kg, v=2200m/s → λ=h/(mv)
Result: λ≈1.80×10−10m = 0.180nm
This is comparable with atomic spacings, which makes neutrons useful for diffraction studies of materials.
Example 4: Macroscopic baseball
m=0.145kg, v=40m/s → λ=6.626×10−34/(0.145×40)
Result: λ≈1.14×10−34m
The wavelength is so small that ordinary diffraction of the baseball is experimentally irrelevant.

Common Mistakes

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Forgetting that the denominator is momentum

The most general relation is λ=h/p. Using h/m or h/v alone is dimensionally wrong and removes one of the quantities that sets momentum.

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Using h/(mv) at highly relativistic speed

When v is no longer much smaller than c, use relativistic momentum p=γmv. The classical expression then overestimates the wavelength.

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Confusing de Broglie wavelength with particle size

The wavelength describes the particle's quantum phase behavior; it is not a physical diameter. An electron can have different de Broglie wavelengths at different momenta without changing its intrinsic identity.

Frequently Asked Questions

What is the de Broglie wavelength?
Every particle has an associated wavelength λ = h/mv where h is Planck's constant. This explains electron diffraction and the wave-particle duality of matter.
Why is it only measurable for small particles?
For a 1 kg ball at 1 m/s, λ = 6.6×10⁻³⁴ m — far smaller than any detector can measure. Only for electrons, neutrons, and atoms is the wavelength large enough to observe.
Does a stationary particle have an infinite de Broglie wavelength?
The ideal formula λ=h/p tends toward infinity as a precisely defined momentum approaches zero. A real localized quantum state is described by a wave packet containing a range of momenta, so the physical interpretation is subtler than assigning one infinite plane-wave wavelength.
Why do electrons diffract from crystals?
Typical electron wavelengths can be similar to the spacing between atomic planes in a crystal. Waves scattered from different planes then interfere constructively at particular angles, producing diffraction peaks analogous to X-ray diffraction and directly demonstrating the wave character of electrons.
How does accelerating voltage affect an electron wavelength?
A larger accelerating voltage gives the electron more kinetic energy and momentum, so its de Broglie wavelength decreases. At modest voltages a nonrelativistic energy-momentum relation is adequate; at high accelerating voltages, relativistic corrections become increasingly important.
Can large objects have de Broglie wavelengths?
Yes. The relation applies to all matter, but macroscopic masses usually have enormous momenta compared with microscopic particles, producing extraordinarily small wavelengths. Quantum interference then becomes exceedingly difficult to isolate from environmental interactions and measurement limits.

Formula Explorer connections

Interpretation: This relationship connects frequency, wavelength, speed, phase, intensity or resonance in an oscillating system. Assumption: Identify the medium, boundary conditions and reference frame. Linear waves, small amplitudes, nondispersive media or ideal resonance may be assumed.

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